It is clear from the definition that PD1 +PD2 ⊆ PD1+D2. We now focus on proving
the other containment. Note that we can construct the Cartier data for each of the above divisors. For everyσ∈Σmax, there existsmσ, m0σ ∈M so thathmσ, uρi=−aρ
and hm0σ, uρi = −bρ for all ρ ∈ σ(1). The sets of all {mσ}σ and {m0σ}σ give
the Cartier data for D1 and D2 respectively. Note the mσ and m0σ are unique
as σ⊥ = {0} for all σ ∈ Σmax. Then the Cartier data for D1 +D2 is precisely
{mσ +m0σ}σ. Now, by Theorem 6.1.7 of [19], PD1 = Conv(mσ|σ ∈ Σ(n)), PD2 =
Conv(m0σ|σ ∈ Σ(n)), and PD1+D2 = Conv(mσ +m
0
σ|σ ∈ Σ(n)). This implies that
for any m ∈ PD1+D2, there are tσ ∈ R≥0 such that P σ∈Σmaxtσ = 1 and m = P σ∈Σmaxtσ(mσ +m 0 σ) = P σ∈Σmaxtσmσ+ P σ∈Σmaxtσm 0 σ ∈PD1 +PD2.
Definition 5.4.4. LetP1, . . . , Pk ⊆MR be lattice polytopes. The Cayley polytope
P1∗. . .∗Pk is defined as
conv(P1× {e1}, . . . , Pk× {ek})⊆MR⊕R
k.
Proposition 5.4.5. Let Σ be a complete fan in NR. Let −Di be nef. Take the
Cayley polytope P−D1 ∗ · · · ∗P−Dk := Conv(P−Di +ei)
k
i=1 ⊆ MR ⊕(R
k)∗. Then
|(ΣE)∨|=Cone(P−D1 ∗ · · · ∗P−Dk).
Proof. We first prove that Cone(P−D1∗· · ·∗P−Dk)⊆ |(ΣE)
∨|. Take an element of the
Cayley polytope, namely, (Pk
i=1tiui, t1, . . . , tk)∈P−D1∗· · ·∗P−Dk, whereui ∈P−Di
and P
iti = 1 where ti ∈R≥0 for all i. For any (v, ϕD1(u) +a1, . . . , ϕDk(u) +ak)∈ |ΣE|Then we can see that
h( k X i=1 tiui, t1, . . . , tk),(v, ϕD1(v) +a1, . . . , ϕDk(v) +ak)i = k X i=1 tihui, vi+ X i=1 ti(ϕDi(v) +ai) ≥ k X i=1 tiai ≥0. (5.4.3)
We just need to prove that|(ΣE)∨| ⊆Cone(P−D1 ∗ · · · ∗P−Dk)).
Suppose (m, t1, . . . , tk)∈ |(ΣE)∨| ∩(MQ⊕Q
k). Then, for allv ∈ |Σ|,
h(m, t1, . . . , tk),(v, ϕ−D1(v), . . . , ϕ−Dk(v)) =hm, vi −
k
X
i=1
tiϕ−Di(v)≥0. (5.4.4)
Then, take T to be the smallest integer so that T ti ∈ Z for all i. Then we just
T m ∈ P−T t1D1−···−T tkDk. As −Di is nef, −T tiDi is nef, hence by the previous
lemma T m ∈ T t1P−D1 + · · ·+ T tkP−Dk. By Proposition 4.3, this implies that
m ∈t1P−D1 +· · ·+tkP−Dk, which proves the desired containment.
5.4.2
Nef Partitions and Batyrev-Borisov Duality
Nef-partitions were a concept that Borisov introduced in order to understand how to look at Calabi-Yau complete intersections in toric fano varieties.
Definition 5.4.6. Let X be a Gorenstein toric Fano variety. A nef partition is
a partition of the torus-invariant prime divisors of X into effective, nef, Cartier divisors D1, . . . , Dr. In other words,
−KX =D1+. . .+Dr.
The associated generic anticanonical complete intersection in the crepant resolution of X is a (possibly singular) Calabi-Yau.
Now to find the polytope equivalent, we first look at the fan. LetDρdenote the
toric divisor associated to ρ ∈ ΣN(∆)(1), where ΣN(∆) is the normal fan of ∆. We then partition ΣN(∆)(1) so that ΣN(∆)(1) =I1∪. . .∪Ik into k disjoint subsets, we
get the divisors Ej :=Pρ∈IjDρ.
Definition 5.4.7. We say the decomposition ΣN(∆)(1) = I1 ∪ . . .∪Ik is a nef-
Note that we can now associate the nef-partition to Minkowski sums of poly- topes. Suppose D is a divisor of the form D = P
ρ∈Σ(1)aρDρ. Take PEi to be the
lattice polytope associated to the divisor Ei. Then:
∆ =P−KX =PE1 +. . .+PEk reflexive.
By abuse of notation, −KX = PEi and ∆ = PPi are both also called nef-
partitions.
We can construct a dual polytope toPi,Qi, as follows:
Qi :={y∈NR :hPj, yi ≥ −δij for all j = 1, . . . , k}
for all i.
Define ∇:= Q1+. . .+Qk. Note that ∇ is reflexive, ∇∗ = conv(P1, . . . , Pk) is
reflexive. Analogously, ∆∗ = conv(Q1, . . . , Qk). Note that hPi, Qji ≥ −δij.
We say a nef-partition is proper if dimPi >0 for alli. Note that any nef-partition
can be reduced to a proper nef partition. We now assume that the nef-partition is proper.
Fix a nef partition Ej of length k of nef divisors and now look at the vector
bundle
E =O(−E1)⊕ · · · ⊕ O(−Ek). (5.4.5)
Then take Θ = (ΣN(∆))E, which is a fan in (N ⊕Zk)⊗R. Then, as Σ is complete
and Ej for all j is nef, then |Θ| is convex, hence we have a dual fan Θ∨. Note that
dual cone to Θ∨, one can then see that|(Θ∨)∨|is the cone over the Cayley polytope
Q1∗ · · · ∗Qk, say σQ [4]. This makes the duality
σP Batyrev-Borisov ←→ σQ the same as ((ΣN(∆))E)∨ ←→(ΣN(∆))E in our context.
Moreover, one can take deg = e1+. . .+ek, deg∨ = e∗1+. . . e
∗
k. We now star
subdivide (((ΣN(∆))E)∨)∨ and ((ΣN(∆))E)∨ by the rays ei and e∗i respectively. Note
that when one does this, the resulting fans are of toric vector bundles E and a nef partition vector bundle of the canonical divisor of the toric variety has polytope ∇. One can now take a A-triangulation of the polytopes Pi and a B-triangulation
of the polytopes Qi in order to give a maximally partial crepant (MPCP) desingu-
larization (as explained in Section 2, or see [2]). This makes the mirror duality
((ΣN(∆))E)∨B ←→((ΣN(∆))E)A as proposed by Borisov.